To determine the time it takes for the penny to hit the ground, we'll use the given height function:
[tex]\[ h(t) = -16t^2 + \text{initial height} \][/tex]
In this scenario, the initial height of the house is 16 feet. Thus, the height function becomes:
[tex]\[ h(t) = -16t^2 + 16 \][/tex]
We need to find the time [tex]\( t \)[/tex] when the penny hits the ground. When the penny hits the ground, the height [tex]\( h(t) \)[/tex] is 0. So we set the equation equal to 0 and solve for [tex]\( t \)[/tex]:
[tex]\[ 0 = -16t^2 + 16 \][/tex]
Firstly, we move the constant term to the other side of the equation:
[tex]\[ -16t^2 + 16 = 0 \][/tex]
[tex]\[ -16t^2 = -16 \][/tex]
To isolate [tex]\( t \)[/tex], we divide both sides by -16:
[tex]\[ t^2 = \frac{-16}{-16} \][/tex]
[tex]\[ t^2 = 1 \][/tex]
Next, we solve for [tex]\( t \)[/tex] by taking the square root of both sides of the equation:
[tex]\[ t = \sqrt{1} \][/tex]
The square root of 1 is 1, so:
[tex]\[ t = 1 \][/tex]
Therefore, it takes 1 second for the penny to hit the ground.
So, the time it takes for the penny to hit the ground is:
[tex]\[ \boxed{1} \text{ seconds} \][/tex]
